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Likability... wow, that's a really broad topic there.
A lot of people have hit on some excellent elements, so I'm going to aim for something else here: believability. And yeah, I'm talking about believability within shows that have talking animals and giant cyborg warriors - if a character doesn't fit with the show's internal logic, things just fall apart, unless it's deliberate and played for laughs.
An example: Azula.
She's a crazy, controlling, psychotic bitch. Loathesome and villainous to the core. So why do I like her? Several reasons. First, she's not one-dimensional. Like 90% of the Avatarverse, there's a deeper story there, one you only get a glimpse of - her father nurtured her sociopathic urges while her mother tried to suppress them. Her actions, her need for control and constant pushing on the boundaries of relationships with her friends, her superiority complex - and the little things like her word choice and mannerisms - they all added up for a deep, psychologically complex, and believable character.
On the flipside, there's Sokka and Toph. Deep? Well, yes, but those depths were largely explored within the series, by virtue of them being the protagonists, but what truly made them believable was the interplay between each other and the rest of the characters. Sure, they were kids who were fighting a war unmatched in history, but they were still kids. The writers never forgot that.